Abstract
Let F be an (n − 1)-dimensional surface of class C 2 in the space ℝn. Given a point p ∈ F, the principal curvatures k 1, k 2,..., k n−1 of F are defined at p. A point p is called umbilical if all principal curvatures at p coincide. The classical Darboux theorem asserts that a surface with every point umbilical is part of a sphere or plane. In the elementary courses of differential geometry, one usually considers only the case n = 3; however, the arguments involved there obviously extend to the case of an arbitrary n. Also, observe that the classical proofs of the Darboux theorem relate to the surfaces of class C 3.
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© 1994 Springer Science+Business Media Dordrecht
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Reshetnyak, Y.G. (1994). Stability in Darboux’s Theorem. In: Stability Theorems in Geometry and Analysis. Mathematics and Its Applications, vol 304. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8360-2_6
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DOI: https://doi.org/10.1007/978-94-015-8360-2_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4467-9
Online ISBN: 978-94-015-8360-2
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